philosophy - logic (Natural Deduction in Propositional Logic)

Question

If astronauts spend long periods in zero gravity only if calcium is reabsorbed in their bodies, then astronauts on a Mars voyage will arrive with brittle bones. If astronauts attempt a voyage to Mars only if they spend long periods in zero gravity, then astronauts on a Mars voyage will arrive with brittle bones. Thus, astronauts on a Mars voyage will arrive with brittle bones. (Z, C, B, V)

P.S: indirect proof(IP) cannot be used! 

- suggest using Reductio ad absurdum and Conditional Proof if needed 

Answer 1

Given:

(I) (Z --> C) --> (V and B)

(II) (V --> Z) --> (V and B)

Prove: V and B

if V, then per (II), Z must hold.

THen per (I), C must hold.

So in this case, V --> Z --> C --> (V and B)

By implication indentity of (2):

not (V --> Z) or (V and B)

Implication property again:

not (not V or Z) or (V and B)

DeMorgans:

(V and not Z) or (V and B) <--- ALPHA

so if Z, then (V and not Z) does NOT hold,

and then per ALPHA, V and B by conjunction